24-cell in nLab

Contents

Contents

Idea

…one of the regular polytopes in dimension 4…

…hence a higher dimensional analog of the Platonic solids…

Definition

The 24-cell is the regular polyhedron in the Cartesian space/Euclidean space ℝ 4\mathbb{R}^4 whose vertices are, under the identification ℝ 4≃ ℝℚ\mathbb{R}^4 \simeq_{\mathbb{R}} \mathbb{Q} with the space of quaternions, the 8 unit quaternions ±1\pm 1, ±i\pm i, ±j\pm j, ±k\pm k and the 16 unit quaternions given by 12(ε 01+ε 1i+ε 2j+ε 3k)\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k) where (ε 0,…,ε 3)∈{−1,1} 4(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4.

(These 24 quaternions form a group under quaternion multiplication, and this group is isomorphic to the binary tetrahedral group.)

Properties

Symmetry group

The finite rotation group inside O(4) which is the symmetry group of the 24-cell is the Coxeter group F4.

• 120-cell

• 600-cell

References

See also

• Wikipedia, 24-cell